All of mine went through a peel off the legs stage, all have grown out of it in different ways. The eldest was in nursery full time same place right through, the worst age was 2-3, slowly started to improve from there, went to school age 5 without a backward glance, loved grannies, visiting friends and sleepovers and is now very confident teenager in some ways,eg first to stand up and make an off the cuff speech in class, but last to join a new club! just doesn't like being thrust into new situations.
DC#2 went part time to 3 different playgroups, from 18m-4 would not leave my lap,finally I got fed up when DC about 3, she was fine about 5 min after I left, but the daily crying went on for months , very wearing. She went to school age 4.10 and loved it at first, then reverted to hating it and crying every morning(although not bullied and fine in class). She changed schools to be with a friend and has never looked back, according to her teacher he has to remind her who the teacher is! Loves trying new things (wanted to try rugby!) but she can still be a whiny and miserable sort of person, so maybe clinginess can just be a more difficult personality, just have to get on with it and not pander to it as can't change it or parent it differently.
DC#3 goes to nursery p/t , got really attached to a staff member when 2 and when she left really went off the rails behaviour wise. For 2 years he clung, cried, tantrumed and clearly stated he hated nursery. When left quickly and cheerfully by a mum who by now had some experience of not pandering to clingy children, he would stand in the corner and not join in. But, he always had a good day playing and joining in when I was not there, according to staff. Age now 4.6, is slowly starting to come out of it, more good days than bad, I have high hopes for school transition not being an issue.
So you are doing the right thing, and he will get better. Its not wrong to cling and cry, he loves you more than anything else!.. and is only just learning the rest of the world can fill his needs too.
Most kids his age do not understand time. Is he happy to be told -sleepover at grans? tell him as late as possible in a simple matter of fact way what is happening, and when time comes to leave do as you do at nursery, be bright and breezy and ensure gran or grandpa has something ready he loves to do. Don't be too specific. I never made too big a deal about coming back either, stayed for a while talking and returned home in leisurely fashion.
Enjoy your time away you lucky thing!